sage: 2+2
4
sage: k=GF(17)
sage: k
Finite Field of size 17
sage: 10+15
25
sage: k(10+15)
8
sage: x=k(2)
sage: y=k(5)
sage: x+y*y
10
sage: P=[x,y]
sage: P
[2, 5]
sage: d=(x^2+y^2-1)/(x^2*y^2)
sage: d
3
sage: d^8
16
sage: def oncurve(P):
....:     x1=P[0]
....:     y1=P[1]
....:     return x1^2+y1^2 == 1 + d*x1^2*y1^2
....: 
sage: oncurve(P)
True
sage: oncurve([3,4])
True
sage: oncurve([3,5])
False
sage: oncurve([3,6])
False
sage: oncurve([3,7])
False
sage: def ed(P1,P2):
....:     x1=P1[0]
....:     y1=P1[1]
....:     x2=P2[0]
....:     y2=P2[1]
....:     x3=(x1*y2+x2*y1)/(1+d*x1*x2*y1*y2)
....:     y3=(y1*y2-x1*x2)/(1-d*x1*x2*y1*y2)
....:     return [x3,y3]
....: 
sage: ed(P,P)
[13, 3]
sage: P2=ed(P,P)
sage: oncurve(P2)
True
sage: P3=ed(P2,P)
sage: P4=ed(P3,P)
sage: P5=ed(P4,P)
sage: P6=ed(P5,P)
sage: P
[2, 5]
sage: P2
[13, 3]
sage: P3
[7, 10]
sage: P4
[14, 4]
sage: P5
[12, 15]
sage: P6
[16, 0]
sage: ed(P6,P6)
[0, 16]
sage: P12=ed(P6,P6)
sage: P24=ed(P12,P12)
sage: P24
[0, 1]
sage: ed(P2,P3)
[12, 15]
sage: P5
[12, 15]
sage: ed(P3,P3)
[16, 0]
sage: